The likelihood functions for the Cauchy maximum likelihood
estimates are given in chapter 16 of
Johnson, Kotz, and Balakrishnan.
These equations typically must be solved numerically on a computer.

Comments

The Cauchy distribution is important as an example of a
pathological case. Cauchy distributions look similar to a normal
distribution. However, they have much heavier tails.
When studying hypothesis tests that assume normality, seeing how the
tests perform on data from a Cauchy distribution is a good indicator
of how sensitive the tests are to heavy-tail departures from
normality. Likewise, it is a good check for robust techniques
that are designed to work well under a wide variety of
distributional assumptions.

The mean and standard deviation of the Cauchy distribution are
undefined. The practical meaning of this is that collecting
1,000 data points gives no more accurate an estimate of the
mean and standard deviation than does a single point.

Software

Many general purpose statistical software programs support at least
some of the probability functions for the Cauchy distribution.